The never ending Golden Fibonacci Fraction
1
Calculate 1 
1
1
1
1
1
1
1  ...
Seems a little overwhelming. Let’s start the process...one step at a time.
1
1  11
1
1
1
1
1
1
 1
= 1
=2
1
11
1
1
1
1
11
1 3
=
2 2
 1
1
= 1
=
1
1
1
1
1
1
 1
1
1
11
= 1
1
5
3
= 1
3
5
=
1
3
2
1
1
2
1
2
 1
3
3
2
5
3
Do you see the substitution.......
Look carefully...just replace the messy fraction portion with
the reciprocal of the previous fraction.
8
5
Calculate decimal values for each fraction and do several more terms. You will see a recognizable quantity.
Keep doing some “donkey” work and it will pay off.
The terms turn out to be the fractions of consecutive terms of the Fibonacci sequence.
Also recall that this gives us the limit of the ratio of the Fibonacci sequence which is the value of the Golden
Ratio, which is
1 5
2
1.618 .
It turns out that some expression which was completely foreign looking is actually a very tidy concept
which turns out to include ideas that didn’t seem to be related.....but in fact are related.
There are many such patterns and concepts that follow these ideas and only through repeated investigation....
we make interesting and important discoveries.